Previous work [1]} set up the concept of spin ice on a general graph. Spin ice concepts are often translatable into the language of graph theory, and vice versa. For instance in the mathematical literature a balanced graph (with even degree nodes) is a directed graph whose indegrees is equal to the outdegrees [2]}. In spin ice language, this corresponds to a configuration of the ice manifold [1]}. Thus, the problem of finding the Pauling entropy [4]} of a spin ice graph is equivalent to the problem of counting the number of balanced digraphs. Moreover, it is one of the many celebrated results by Euler that only graphs which can be balanced via an orientation support an Eulerian trail [5]}.
In this sense, many results from graph theory can be borrowed to study frustration in spin ice.
